Calculating the Area Enclosed by the Graphs of y3/x and y4-x

In this article, we will explore the process of calculating the area enclosed by the graphs of y3/x and y4-x. This involves understanding where these two functions intersect, determining which function is greater over the interval of intersection, and integrating the difference between the functions to find the enclosed area. This process is a fundamental application of calculus and integration techniques.

Identifying Intersection Points

The first step in solving this problem is to determine the points of intersection between the two functions. We do this by setting the equations equal to each other:

3/x 4 - x

Multiplying through by x to clear the fraction, we get:

3 4x - x^2

Which simplifies to the quadratic equation:

x^2 - 4x 3 0

Solving this quadratic equation, we find the roots:

x^2 - 3x - x 3 0

x(x - 3) - 1(x - 3) 0

(x - 1)(x - 3) 0

Therefore, the solutions are:

x_1 1 and x_2 3

Determining the Function Gaps

Next, we need to determine which function is larger over the interval [1, 3]. We can choose two points within the interval to check. For example, at x 2:

f(2) 3/2 1.5

g(2) 4 - 2 2

Since 1.5 4 - x is greater than 3/x.

Setting Up the Integral

Given that 4 - x is the larger function over the interval [1, 3], the area between the curves is given by the integral of the difference between the two functions:

A ∫[1 to 3] (4 - x - 3/x) dx

Evaluating the Integral

Breaking down the integral:

A ∫[1 to 3] 4 - x dx - ∫[1 to 3] 3/x dx

We integrate each term separately:

A [4x - (x^2)/2]_1^3 - 3 [ln|x|]_1^3

Evaluating these at the upper and lower limits:

A [4(3) - (3^2)/2] - [4(1) - (1^2)/2] - 3 [ln(3) - ln(1)]

A [12 - 9/2] - [4 - 1/2] - 3 [ln(3) - 0]

A 27/2 - 7/2 - 3 ln(3)

A 10 - 3 ln(3)

The exact area enclosed by the graphs of y3/x and y4-x is approximately 0.704 square units.

Note: The final area calculated is approximately 10 - 3 ln(3).

Understanding the intersection points and the process of integration are crucial steps in solving such problems. This method can be extended to other similar problems involving curves in calculus.